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Basis number of bounded genus graphs

Florian Lehner, Babak Miraftab

Abstract

The basis number of a graph $G$ is the smallest integer $k$ such that $G$ admits a basis $B$ for its cycle space, where each edge of $G$ belongs to at most $k$ members of $B$. In this note, we show that every non-planar graph that can be embedded on a surface with Euler characteristic $0$ has a basis number of exactly $3$, proving a conjecture of Schmeichel from 1981. Additionally, we show that any graph embedded on a surface $Σ$ (whether orientable or non-orientable) of genus $g$ has a basis number of $O(\log^2 g)$.

Basis number of bounded genus graphs

Abstract

The basis number of a graph is the smallest integer such that admits a basis for its cycle space, where each edge of belongs to at most members of . In this note, we show that every non-planar graph that can be embedded on a surface with Euler characteristic has a basis number of exactly , proving a conjecture of Schmeichel from 1981. Additionally, we show that any graph embedded on a surface (whether orientable or non-orientable) of genus has a basis number of .

Paper Structure

This paper contains 4 sections, 9 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basis number of graphs embedded on surfaces with low genus
  4. Basis numbers of graphs on surfaces of higher genus

Key Result

Lemma 1

A graph is planar if and only if there is a basis for its cycle space such that every edge lies in at most two elements of the basis.

Figures (2)

  • Figure 1: Fundamental polygons obtained by cutting along the edges of $x \cup y$. The left side results from cutting the torus, the right side from cutting the Klein bottle.
  • Figure 2: A path $Q$ separating the fundamental polygon into an area below $Q$ and an area above $Q$

Theorems & Definitions (15)

  • Lemma 1: MacLane’s planarity criterion maclane1970combinatorial
  • Definition 1: cf.MR615307
  • Conjecture 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • proof
  • Theorem 6
  • proof
  • ...and 5 more